# General solution to homogeneous difference equation

With a given example

$$a_{n-1} = ca_{n-2}$$

general solution:

$$a_{n} = c . c . a_{n-2}$$
$$= c . c . a_{n-3}$$
$$= c^n a_0$$

Question: Find the general solution for the homogeneous equation $$a_{n} = 5a_{n-1}$$

General solution:

$$a_{n} = 5^n a_{0}$$

Is my general solution correct based on the given example?

• I believe you have a typo in the first equation, should it be $a_n = c \cdot a_{n-1}$? – Nicolas Villanueva May 5 '11 at 7:59
• Yes. There is a typo in the first displayed formula, you meant $a_n=c\cdot a_{n-1}$, and in the third displayed line, you meant $a_n=c\cdot c\cdot c\cdot a_{n-3}$. And I would prefer to go forwards than backwards, $a_1=5a_0$, $a_2=5a_1=5^2a_0$, and so on. But backwards is fine. – André Nicolas May 5 '11 at 8:00
• @Nicolas Villanueva: I have corrected the typo. @yunone: I am new to mathse. Thanks for the info. I will accept the answers – ilovetolearn May 5 '11 at 8:10
• @liangneh: The correction of the first typo should have been to $a_n=c\cdot a_{n-1}$. The correction you made is correct, but not as helpful. The next typo, on the third displayed line, remains uncorrected as of now. – André Nicolas May 5 '11 at 8:37

Even if that first equation is a typo, that is the correct general solution. If a proof is necessary, I suggest using Induction on $n$.